1898,] with two-dimensional fluid motion. 387 



the singular points lie within the circles 0/, C 1 , ..., C p , C p , and 

 consist of pairs of points inverse to one another in regard to C . 

 This is clear either because the substitution S> is equivalent to an 

 inversion in regard to the circle C followed by an inversion in 

 regard to the circle G r , or, what is the same thing, by remarking 

 that if 



</>=... £»'^ ... , <//=... v ?1 ^ "■ • ■ • 



be two substitutions in which the primary factors occur in the 

 same order but are the inverses of one another, and, if z, z be 

 conjugate complexes, then also </> (z) and <}>' (z') are conjugate 

 complexes. But in the case in which the circles G , (7/, ..., G p are 

 all cut at right angles by another circle, 0, it follows further that 

 the singular points all lie upon this circle 0. 



Hence follows immediately another consequence ; we have 

 already seen that the imaginary part of the function 



is constant on each of the circles (7/, ..., G p ', C , being zero on C ; 

 it can now be seen that the real part of v$ n is constant on each of 

 the arcs of the orthogonal circle which are limited by the circles 

 (7/, . . . , Gp, G lt ..., G p . For if j, j' denote two substitutions such as 

 (/> and </>', we can put 



and ^j (B n ), Sy (A n ) — as also %< (B n ), ^ (A n ) — will be two singular 

 points having conjugate complex positions upon 0. As the real 

 part of v^ depends only on the arguments of the factors 



[?-M^)]/[r-*M^)i 



the result follows at once. 



From this, if barriers be drawn in O, consisting of the p straight 

 lines joining G 1 'G 1 , G 2 'G. iy ..., G p 'G p , we find the equations 

 v a n ,c n = -.± } or q, as n = r, or n^r. 



It will be sufficient to consider one case. Let ZY be the inverse 

 point of (7/ in regard to the circle 0, and let K denote the real 

 part of v^: a at the point Oj ; then since the function —iv$ r is real 

 on the arc ac x it has conjugate complex values along the arcs cfJ^, 

 CxDi ; so that we have on these arcs, respectively, 



